Numerical Integration Issues

During the development of several simulation studies using the current Aspen Version 7.3, a strange low-amplitude oscillation has been experienced, which is not affected by controller tuning. The solution to the problem was found to be a modification in the integrator. 

Aspen users should be aware of this problem. 

The heterogeneous azeotropic distillation process provides an excellent example of the utility of distillation simulation to both design and control of a very complex nonideal system. 

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Computational issues that are pertinent in MD simulations are time complexity of the force calculations and the accuracy of the particle trajectories including other necessary quantitative measures. These two issues overwhelm computational scientists in several ways. MD simulations are done for long time periods and since numerical integration techniques involve discretization errors and stability restrictions which when not put in check, may corrupt the numerical solutions in such a way that they do not have any meaning and therefore, no useful inferences can be drawn from them. Different strategies such as globally stable numerical integrators and multiple time steps implementations have been used in this respect (see [27, 31]). 

Except for occasional discussions of the basis set dependence of the results, the numerical implementation issues such as grid integration techniques, electron-density fitting, frozen-cores, pseudopotentials, and linear-scaling techniques, are omitted. 

The oxidative deterioration of most commercial polymers when exposed to sunlight has restricted their use in outdoor applications. A novel approach to the problem of predicting 20-year performance for such materials in solar photovoltaic devices has been developed in our laboratories. The process of photooxidation has been described by a qualitative model, in terms of elementary reactions with corresponding rates. A numerical integration procedure on the computer provides the predicted values of all species concentration terms over time, without any further assumptions. In principle, once the model has been verified with experimental data from accelerated and/or outdoor exposures of appropriate materials, we can have some confidence in the necessary numerical extrapolation of the solutions to very extended time periods. Moreover, manipulation of this computer model affords a novel and relatively simple means of testing common theories related to photooxidation and stabilization. The computations are derived from a chosen input block based on the literature where data are available and on experience gained from other studies of polymer photochemical reactions. Despite the problems associated with a somewhat arbitrary choice of rate constants for certain reactions, it is hoped that the study can unravel some of the complexity of the process, resolve some of the contentious issues and point the way for further experimentation. 

In its simplest sense classical molecular dynamics involves integrating the classical equations of motion for a set of molecules. In this regard, the most salient issues concern initial simulation conditions, numeric integration of the equations of motion, ensembles selection, equilibration, and checks on the simulation. 

Given the appropriate potential energy diagrams from the DLVO theory, the stability ratio may be calculated by graphical or numerical integration and then compared with experimental values of W=kyk, the ratio of the experimental rate constants for rapid and slow flocculation. Such a comparison is a severe test of the applicability of theory to experiment, and the observed deviations, although often not appreciable, reflect the assumptions and approximations which are necessary in the calculation of the potential energy terms. An advanced treatment of these issues will be found in Russel et al.- . 

In this section we discuss the issues of convergence and accuracy in numerical integration methods for solving ordinary differential equations. 

There were no dynamic simulation issues experienced in this system. The default Implicit Euler numerical integration algorithm worked well, giving quite short simulation times (1 min of real time to simulate lOh of process time). 

One other feature of DFT calculations worth mentioning concerns the manner in which the term is calculated. While all the other terms are evaluated in a way similar to HF theory, cannot be evaluated analytically. Instead, a numerical integration is performed to evaluate Basically, a grid of points is placed around the molecule, and E - is evaluated at each point. A key issue is how fine this grid is. More points will produce better results, but at a cost of more computing time. A balance must be struck, therefore, and this is something the user needs to keep in mind. 

B. Simeon, An Extended Descriptor Form for the Numerical Integration of Multibody Systems, J. Appl. Numer. Math., Special Issue on the Proceedings of the NUMDIFF 6 Conference, Halle, September 1992. 

In addition we should also note that the aforementioned sensitivity loss is not the only issue here. It is also clearly visible in Figure 7(c) that the shape of the ring current baseline is not very well-defined. Several difficulties may arise from this fact, especially if we would like to determine the amount of the collected intermediates by the means of numerical integration. 

On the right-hand side of the constitutive equation, Eq. (1.3), a diffusion term has been added, as proposed by Sureshkumar and Beris [81], so that in turbulent simulations the high wavenumber contributions of the conformation tensor do not diverge during the numerical integration of this equation in time. This parallels the introduction of a numerical diffusion term in any scalar advection equation (e.g., a concentration equation with negligible molecular diffusion) that is solved along with the flow equations under turbulent conditions [82]. In Eq. (1.3), Dq is the dimensionless numerical diffusivity [54-56]. The issue of the numerical diffusivity is further discussed in Sections 1.3.2 and 1.4.3. 

Unlike FEP, there is no finite difference term in the quantity whose ensemble average we must determine, so differences between the potential surfaces of neighboring X states are not explicitly an issue. Instead, for TI the main concern with respect to the X pathway is that we select enough X points so that the numerical integration over these points is reasonably accurate. For smoothly and slowly varying AG versus X curves, a modest number of points will usually suffice. Fortunately, such a curve is characteristic of most free energy calculations. 

The only issue left is that O Eq. 25.24 cannot generally be solved algebraically. Instead a numerical integration scheme (such as Gauss Quadrature) has to be used to determine the stillness matrix. An integral having arbitrary limits can be transformed so that its limits are from — 1 to 1, e.g. ... 

Following a discussion of polynomial interpolation and numerical integration, a survey is presented of the major techniques for solving IVPs, as implemented in MATLAB. Then, the issues of numerical accuracy and stability are treated at deptii for commonly-used ODE solvers. Next, we consider differential-algebraic equation (DAE) systems that contain both... 

The many approaches to the challenging timestep problem in biomolecular dynamics have achieved success with similar final schemes. However, the individual routes taken to produce these methods — via implicit integration, harmonic approximation, other separating frameworks, and/or force splitting into frequency classes — have been quite different. Each path has encountered different problems along the way which only increased our understanding of the numerical, computational, and accuracy issues involved. This contribution reported on our experiences in this quest. LN has its roots in LIN, which... 

System integration involves numerous miscellaneous development activities, such as control software to address system start-up, shut-down and transient operation, and thermal sub-systems to accomplish heat recovei y, heat rejection and water recoveiy within the constraints of weight, size, capital and operating costs, reliability, and so on. Depending on the application, there will be additional key issues automotive applications, for example, demand robustness to vibrations, impact, and cold temperatures, since if the water freezes it will halt fuel cell operation. 

For models described by a set of ordinary differential equations there are a few modifications we may consider implementing that enhance the performance (robustness) of the Gauss-Newton method. The issues that one needs to address more carefully are (i) numerical instability during the integration of the state and sensitivity equations, (ii) ways to enlarge the region of convergence. 

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